Device for the graphical solution of simultaneous equations

ABSTRACT

A device for the graphical solution of the simultaneous equations A f (D, C, K), which equations can be represented by nomographs, e.g., Larson&#39;&#39;s Nomograph, having two spaced vertical axes bounding two families of curves. The device comprises a first straight line defining means (e.g., a transparent stick) having one end thereof presettable along the first vertical axis, a second straight line defining means (e.g., a second transparent stick) having one end thereof also presettable along the first vertical axis, their other ends intersecting at a common point on the second vertical axis, and means (e.g., a pin) disposed at the common point and movable along the second vertical axis.

United States Patent Ladany s4] DEVICE FOR THE GRAPHICAL SOLUTION 0F SIMULTANEOUS EQUATIONS [72] Inventor: Shaul Ladany, Ramat Efal, Rechov Haelah 6, Doar Ramat Gan, Israel 1 Aug. 22, 1972 Primary Examiner-Stephen .I. Tomsky [22] Filed: May 20, 1971 Assistant Examiner-Stanley A. Wal [211 App! NOJ 145,391 Attorney-Ben amin J. Bansh ABSTRACT 12%} 12581211313113:311::::::13:::3iiiiiiiiiiiiiiiiiiiieifi iofi A device the graphical e of thefiimultane- [58] Field of SearchW235/61 B 89 R 61 GM 85 ous equations A =f(D, C, K), WhlCll equations can be [1 SD represented by nomographs, e.g., Larsons Nomograph, having two spaced vertical axes bounding two [56] References Cited families of curves. The device comprises a first straight line defining means (e.g., a transparent stick) UNITED STATES PATENTS having one end thereof presettable along the first ver-' tical axis, a second straight line defining means (e.g., a g second transparent stick) having one end thereof also 2520904 9/1950 g e a 235/89 R presettable along the first vertical axis, their other 2 666 577 1/1954 Pgr ke r ends intersecting at a common point on the second 3 044 692 7/ 1962 Buus 2 [61 B vertical axis, and means (e.g., a pin) disposed at the common point and movable along the second vertical FOREIGN PATENTS OR APPLICATIONS I 990,243 6/ 1951 France ..235/61 B 8 Claims, 4 Drawing Figures rmmi'mua lm 3.685.724

' SHEET 2 BF 2 FIG.2 2

Tm Q5 l l a i Q T T! T SHAUL LADANV L ATTORNEY INVENTOR- DEVICE FOR THE GRAPHICAL SOLUTION OF SIMULTANEOUS EQUATIONS BACKGROUND OF THE INVENTION The present invention relates to a device for the graphical solution of simultaneous equations, particularly of equations of the form A f (D, C, K) which can be represented by nomographs having two spaced vertical axes bounding two families of curves, one vertical axis beingrepresentative of A, the other vertical axis being representative of D, while C and K are constants each represented by a family of curves.

One such nomograph is in the field of statistical sampling plans and is called Larsons Nomograph of cumulative binomial distributions. Such nomographs are used for designing single-stage sampling plans. (See Larson, H.R., A Nomograph of the Cumulative Binomial Distribution, Industrial Quality Control, Vol. 23, No. 6, December 1966, pp. 270278.). The invention is particularly useful for designing sampling plans for individual lots when the lot size andsample size are large and the numberof defectives in the lot is relatively small to the size of the lot, using Larsons Nomograph, and is therefore described below with respect to that application for purposes of example.

A single fraction-defective sampling plan for individual lots is designed for the conditions that a lot with a certain fraction defective p will have a probability of rejection or, while giving protection to the consumer such that if the lot fraction defective is p the probability of acceptance shall be [3. Simultaneous solution of the cumulative hypergeometric equations 1 and (2) for n and c will yield the required plan.

n the sample size p N= the acceptable number of defectives in the lot p N= the rejectable number of defectives in the lot x the number of defectives in the sample the acceptance number It should be noted that, because of the discreteness of the system, simultaneous solution of equations (1) and (2) will seldom yield values for n and c that exactly satisfy the values of l a and ,8. Economic considerations will determine whether the risks obtained are satisfactory, or whether a different set of n, c values will be required.

It is not possible to obtain a direct simultaneous solution of equations (1) and (2). A trial-and-error method using tables has been given, but this is somewhat tedious to apply and is limited by the fact that the tables only apply for lot sizes no greater than 100 (with some scattered exceptions).

Although these equations also can only be solved by trial and error, they have long constituted the easiest approach to the design of suchsampling plans.

SUMlVlARY OF THE PRESENT INVENTION The present invention provides a device enablingthe rapid solution of these equations using the Larson Nomograph. 4

Larsons Nomograph of the cumulative binomial distribution is reproduced in FIG. 1. In the usual procedure of using this nomograph with binomial probabilities, the acceptable and rejectable probabilities of success are used directly with the'corresponding. values of l a and [3 to determine the sample size and the acceptance number.

However, in applying the Binomial ll approximation to the hypergeometric distribution it becomes clear that the value taken as the probability of occurrence n/N is unknown, because it is based on the as-yet unknown sample size. Further, the number of trials p N in equation (3) differs from the number of trials p N in equation (4). Nevertheless, in spite of these difficulties, which prevent direct solution of equations (3) and (4), a simple trial-and-error procedure can be used with the Larson Nomograph to yield the desired results.

The device of the present invention enables the desired results to be obtained in a very rapid manner from Larsons Nomograph.

Briefly, the device of the present invention provides means defining the two vertical axes of the Nomograph. The preferred embodiment of the invention utilizes a board having two spaced vertical slots, each including the appropriate scale markings appearing on the two vertical scales of the Nomograph. The device further includes two straight line defining means which, in' the preferred embodiment, are in the form of transparent sticks each having a straight opaque line drawn thereon. One end of each of the sticks is presettable along one of the vertical axes, and the other ends of the two sticks intersect at a common point on the second vertical axis. Means are provided disposed at the common point and movable along the second vertical axis while the first and second line defining means are maintained straight. In the preferred embodiment, this is accomplished by making the other ends of the two sticks bifurcated, and providing a pin within the bifurcations and movable along the slot of the second vertical axis.

The specific procedure in which the device is used for obtaining the desired results from the Larsons Nomograph, is described below with respect to a specific example.

Also. described below are variations in the novel device utilizing a rubber band or a piece of string, rather than the bifurcated sticks of the preferred embodiment.

BRIEF DESCRIPTION OF THE DRAWINGS The invention is herein described more particularly with reference to the drawings, wherein:

FIG. 1 illustrates Larsons Nomograph of the cumulative binomial distribution;

FIG. 2 illustrates a device constructed in accordance with the invention for yielding the desired results from Larsons Nomograph;

FIG. 3 illustrates another form of the present invention, utilizing a rubber band rather than rigid sticks; and

FIG. 4 illustrates a further form, utilizing a piece of string rather than rigid sticks.

DESCRIPTION OF THE PREFERRED EMBODIMENTS The device of FIG. 2 enables one to rapidly obtain 7 the desired results form Larsons Nomograph of FIG. 1. It includes a board 2 formed with a pair of horizontallyspaced vertical slots 4,6, the space 8 between the two slots being adapted to receive Larsons Nomograph, generally designated 10. Vertical slot 4 carries scale markings 12 representing scale A of the Nomograph, and vertical slot 6 carries scale markings 14 representing scale B of the Nomograph.

The device further includes a pair of transparent sticks 16, 18, each having an opaque line 16, 18', running along its longitudinal center. A pin 20 passes through an opening formed in one end of stick l6 and is movable in vertical slot 4, and a second pin 22 passes through an opening formed in the corresponding end of stick 18 and is also movable in vertical slot 4, Each pin carries a pointer 20, 22', cooperable with the scale markings 12.

The opposite ends of sticks 16, 18, are both bifurcated, as shown at 24, 26. These ends are made to intersect at a point on vertical slot 6, and a pin 28 movable in slot 6 passes through the intersecting bifurcated ends of the sticks. Pin 28 carries a pointer 28 cooperable with scale markings l4.

Pins 20, 22 are presettable at predetermined positions in vertical slot 4 at the beginning of the procedure, and once preset they are not intended to be moved during the remainder of the procedure. For this purpose, these pins may be of such dimensions to exert a significant friction when moved in slot 4. Altematively, they may each be formed with (e.g., threaded, or balI-and-detent) retaining elements. Pin 28, however, is intended to be freely movable in slot 6, and is therefore movable therein with a low coefficient of friction.

The device illustrated in FIG. 2 may be used for rapidly obtaining the desired information from Larsons Nomograph of FIG. 1 in the following manner:

In the Binomial II approximation, the probability occurrence is taken as n/N, and the numbers of trials are taken as p N and p N. These points are shown in FIG. 2.

The following example concerns the problem of finding the sample size and acceptance number for sampling from a lot of 200 items, given the following conditions: p 0.02, a 0.05, p, 0.08, and B 0.10. In such case,p N=4,andp N= l6.

Pin 20 is preset in slot 4 to the value of 3, namely 0.10 on scale 12. Pin 22 is preset in slot 4 to the (l a) position, namely 0.95, on scale 12. Pin 28 is then moved along slot 6 until the lines 16' and 18, on sticks 16 and 18, respectively, are placed so that (as nearly as possible):

1. Line 16 intersects the p N= l6 curve;

2. Line 18' passes through p N =4 curve; and

3. The intersecting points on both of the above curves lie on a single C-curve.

When these conditions obtain, the acceptance number c is read directly, and the sample size n is obtained by multiplyingthe value of n/N on the lefthand scale 14 by the lot size N. In the present example, n=(200) (0.295), or 59; and c=2.

FIG. 3 illustrates a device utilizing a rubber band which may be used for following the same procedure as described above. In this variation, the rubber band, generally designated 30, has one end fixed or presettable in a predetermined position along vertical scale 31, e.g., by means of a pin 32, and the opposite end of the rubber band is also fixed or presettable along scale 31, e.g., by a second pin 33. An intermediate portion of rubber band 30 is made to pass around a further pin 34 movable along a second vertical scale 35. The rubber band is tensioned so that when pin 34 is moved the tension will maintain its two portions 36, 38 in straight lines. Portion 36 thus corresponds to straight lines 16' of FIG. 2, and portion 38 corresponds to straight line 18' of FIG. 2.

FIG. 4 illustrates a further variation of the device including a single piece of string. In FIG. 4, one end of the piece of string, generally designated 40, passes around a pin 42 presettable along vertical scale 41, and the opposite end of the string passes around a second pin 43 presettable along the same vertical scale. The intermediate portion of string 40 passes around a further pin 44 movable along vertical scale 45. During this movement of pin 44, tension is maintained in the string, e.g., by the user hand 50, so that string portions 46, 48

are maintained in straight lines and thus serve the same functions as lines l6, 18' of FIG. 2.

Many variations, modifications, and other applications of the illustrated embodiments will be apparent.

What is claimed is:

1. A device for the graphical solution of the simultaneous equations A f (D, C, K), which equations can be represented by nomographs having two spaced vertical axes bounding two families of curves, one vertical axis representing .4, the other vertical axis representing D, while C and K are constants each represented by a means having one end thereof also presettable along said first vertical axis, the other ends of said first and second straight line defining means intersecting at a common point on said second vertical axis, and means disposed at said common point and movable along said second vertical axis while said first and second line defining means are maintained straight during said movement.

2. A device as defined in claim 1, wherein said first and second straight line defining means are each in the form of a rigid, elongated member.

3. a device as defined in claim 2, wherein each of said rigid members is a transparent stick having a straight opaque line drawn thereon.

4. A device as defined in claim 1, wherein each of said first and second straight lines definingmeans is in the form of a rigid stick each pivotably mounted on said one end to a pin presettable along said first vertical axis, the opposite ends of said stick being bifurcated and received on a further pin movable along said second vertical axis.

5. A device as defined in' claim 4, wherein said device includes a board formed with a pair of vertical slots constituting said first and second vertical axes, said first-mentioned pins being presettable along said first vertical slot, and said further pin being movable along said second vertical slot while engaging said bifurcated ends of the sticks. I

6. A device as defined in claim 1, wherein said first and second straight line defining means are constituted of a single rubber band, the opposite ends of which are fixable at any point on said first vertical axis, an intermediate point of the rubber band being passed around an element which is movable along said second vertical axis, the rubber band being tensioned to maintain in straight lines the two portions between said intermediate point and the two ends, as said element is moved along the second verfical axis.

7. A device as defined in claim 1, wherein said first and second straight line defining means are constituted of a single piece of string, an intermediate point of which is passed around an element movable along said second vertical axis, the opposite ends of the string passing around two further elements presettable along said first vertical axis, the extremeties of said opposite ends being tensioned while said first-mentioned element is moved along saidsecond vertical axis.

8. A device as defined in claim 1, which device is used for the graphical solution of Larsons Nomograph of cumulative binomial distributions for designing single-stage attribute plans for individual small lots, the C- family of curves representing the number of occurrences, the K-family of curves representing the number of trials or sample size, said first vertical axis including scale markings representing the probability of C-occurrence or fewer occurrences in K-trials, and said second vertical axis includes scale markings representing the probability of occurrence in a single trial. 

1. A device for the graphical solution of the simultaneous equations A f (D, C, K), which equations can be represented by nomographs having two spaced vertical axes bounding two families of curves, one vertical axis representing A, the other vertical axis representing D, while C and K are constants each represented by a family of curves; the device comprising, means defining the first vertical axis and including scale markings representing the value of A, means defining the second spaced vertical axis and including scale markings representing the value of D, the two families of curves representing the constants C and K being disposable between said two vertical axes, first straight line defining means having one end thereof presettable along said first vertical axis, second straight line defining means having one end thereof also presettable along said first vertical axis, the other ends of said first and second straight line defining means intersecting at a common point on said second vertical axis, and means disposed at said common point and movable along said second vertical axis while said first and second line defining means are maintained straight during said movement.
 2. A device as defined in claim 1, wherein said first and second straight line defining means are each in the form of a rigid, elongated member.
 3. a device as defined in claim 2, wherein each of said rigid members is a transparent stick having a straight opaque line drawn thereon.
 4. A device as defined in claim 1, wherein each of said first and second straight lines defining means is in the form of a rigid stick each pivotably mounted on said one end to a pin presettable along said first vertical axis, the opposite ends of said stick being bifurcated and received on a further pin movable along said second vertical axis.
 5. A device as defined in claim 4, wherein said device includes a board formed with a pair of vertical slots constituting said first and second vertical axes, said first-mentioned pins being presettable along said first vertical slot, and said further pin being movable along said second vertical slot while engaging said bifurcated ends of the sticks.
 6. A device as defined in claim 1, wherein said first and second straight line defining means are constituted of a Single rubber band, the opposite ends of which are fixable at any point on said first vertical axis, an intermediate point of the rubber band being passed around an element which is movable along said second vertical axis, the rubber band being tensioned to maintain in straight lines the two portions between said intermediate point and the two ends, as said element is moved along the second vertical axis.
 7. A device as defined in claim 1, wherein said first and second straight line defining means are constituted of a single piece of string, an intermediate point of which is passed around an element movable along said second vertical axis, the opposite ends of the string passing around two further elements presettable along said first vertical axis, the extremeties of said opposite ends being tensioned while said first-mentioned element is moved along said second vertical axis.
 8. A device as defined in claim 1, which device is used for the graphical solution of Larson''s Nomograph of cumulative binomial distributions for designing single-stage attribute plans for individual small lots, the C-family of curves representing the number of occurrences, the K-family of curves representing the number of trials or sample size, said first vertical axis including scale markings representing the probability of C-occurrence or fewer occurrences in K-trials, and said second vertical axis includes scale markings representing the probability of occurrence in a single trial. 